Spread of a Gaussian Wave Packet with Time

A localized free particle can be represented by a Gaussian wave packet. This Demonstration shows the spreading of a Gaussian wave packet, considering the effects of varying particle mass and momentum and initial width of the wave packet. Choose the parameters: mass, initial width and momentum, and see the evolution of the wave packet with time.

SNAPSHOTS

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DETAILS

A Gaussian wave packet at time can be represented by
,
where is the initial width of the wave packet and is the momentum.
At a later time , the wave packet evolves with
,
where is a constant, is the mass of the particle, and is a parameter that determines the corresponding width of the wave packet ( is the actual width). The parameter is given by
.
Clearly, the width increases with time , as the wave packet spreads. For simplicity, and have been set equal to unity.
The probability amplitude is a Gaussian function centered about the point .
The wave packet maintains a Gaussian shape, with changing centroid and width.
Snapshot 1: small initial width implies faster spread
Snapshot 2: initially broad wave packet spreads out more slowly
In the limiting case, of a wave packet initially equal to a Dirac delta function, it immediately transforms into an infinite plane wave.
Reference
[1] H. C. Verma, Quantum Physics, 2nd ed., Bhopura, Ghaziabad, India: Surya Publications, 2009.
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